Here's what I know about it from installing it several different ways on other distributions.
Of the source packages at www.alsa-project.org, not all are necessary.
From alsa-utils, the important components are alsamixer and alsactl.
Nothing is crucial from alsa-tools.
The alsa-oss library is only necessary to support audio applications which are not alsa-aware. Most applications these days are alsa-aware, and it would be a better policy not to include such applications in Puppy, anyway.
Both Xine and Snack support ALSA, although they might need to be recompiled or reconfigured.

At one stage I copied the ALSA files from a full Debian installation into a small distribution.
The main driver files are at /lib/modules/2.4.xx/kernel/sound/
The alsa library is /usr/lib/libasound.so.xxx
One crucial file is the initialization script - /etc/init.d/alsa, normally run at startup. It creates these devices - /dev/dsp /dev/mixer /dev/sequencer /dev/midi.